A060107 -id:A060107 - cp's OEIS frontend

A083120 Numbers that are congruent to {0, 2, 4, 5, 7, 9, 10} mod 12.

0, 2, 4, 5, 7, 9, 10, 12, 14, 16, 17, 19, 21, 22, 24, 26, 28, 29, 31, 33, 34, 36, 38, 40, 41, 43, 45, 46, 48, 50, 52, 53, 55, 57, 58, 60, 62, 64, 65, 67, 69, 70, 72, 74, 76, 77, 79, 81, 82, 84, 86, 88, 89, 91, 93, 94, 96, 98, 100, 101, 103, 105, 106, 108, 110

Offset: 1

James Ingram (j.ingram(AT)t-online.de), Jun 01 2003

Key-numbers of the pitches of a Mixolydian mode scale on a standard chromatic keyboard, with root = 0. A Mixolydian mode scale can, for example, be played on consecutive white keys of a standard keyboard, starting on the root tone G.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

A guide for some sequences related to modes and chords:
Modes:
Lydian mode (F): A083089
Ionian mode (C): A083026
Mixolydian mode (G): this sequence
Dorian mode (D): A083033
Aeolian mode (A): A060107 (raised seventh: A083028)
Phrygian mode (E): A083034
Locrian mode (B): A082977
Chords:
Major chord: A083030
Minor chord: A083031
Dominant seventh chord: A083032

[n : n in [0..150] | n mod 12 in [0, 2, 4, 5, 7, 9, 10]]; // Wesley Ivan Hurt, Jul 20 2016

A083120:= n-> 12*floor((n-1)/7)+[0, 2, 4, 5, 7, 9, 10][((n-1) mod 7)+1]:
seq(A083120(n), n=1..100); # Wesley Ivan Hurt, Jul 20 2016

Select[Range[0,120], MemberQ[{0,2,4,5,7,9,10}, Mod[#,12]]&] (* Harvey P. Dale, Feb 20 2011 *)
LinearRecurrence[{1, 0, 0, 0, 0, 0, 1, -1}, {0, 2, 4, 5, 7, 9, 10, 12}, 70] (* Jianing Song, Sep 22 2018 *)
Quotient[4 (3 # - 2), 7] & /@ Range[96] (* Federico Provvedi, Nov 06 2023 *)

a(n)=[-2, 0, 2, 4, 5, 7, 9][n%7+1] + n\7*12 \\ Charles R Greathouse IV, Jul 21 2016

my(x='x+O('x^99)); concat(0, Vec(x^2*(2+2*x+x^2+2*x^3+2*x^4+x^5+2*x^6)/((x^6+x^5+x^4+x^3+x^2+x+1)*(x-1)^2))) \\ Jianing Song, Sep 22 2018

G.f.: x^2*(2 + 2*x + x^2 + 2*x^3 + 2*x^4 + x^5 + 2*x^6)/((x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)*(x - 1)^2). - R. J. Mathar, Oct 08 2011
From Wesley Ivan Hurt, Jul 20 2016: (Start)
a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8.
a(n) = (84*n - 77 + 5*(n mod 7) - 2*((n + 1) mod 7) - 2*((n + 2) mod 7) + 5*((n + 3) mod 7) - 2*((n + 4) mod 7) - 2*((n + 5) mod 7) - 2*((n + 6) mod 7))/49.
a(7k) = 12k - 2, a(7k-1) = 12k - 3, a(7k-2) = 12k - 5, a(7k-3) = 12k - 7, a(7k-4) = 12k - 8, a(7k-5) = 12k - 10, a(7k-6) = 12k - 12. (End)
a(n) = a(n-7) + 12 for n > 7. - Jianing Song, Sep 22 2018
a(n) = floor(4 * (3*n - 2) / 7). Federico Provvedi, Nov 06 2023

A059620 Colors of the 88 keys of the standard piano: white keys = 0, black keys = 1, start with A0 = the 0th key.

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 0

Offset: 0

Naohiro Nomoto, Feb 19 2001

			.1..1.1..1.1.1..1.1..1.1.1..1.1..1.1.1..1.1..1.1.1..1.1..1.1.1..1.1..1.1.1..1.1..1.1.1
0.00.0.00.0.0.00.0.00.0.0.00.0.00.0.0.00.0.00.0.0.00.0.00.0.0.00.0.00.0.0.00.0.00.0.0.00

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A060106, A060107, A081031, A081032.

Flatten[Table[{0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1}, {8}]][[1 ;; 88]] (* Jean-François Alcover, Apr 24 2017 *)
PadRight[{},88,{0,1,0,0,1,0,1,0,0,1,0,1}] (* Harvey P. Dale, Sep 14 2020 *)

a(n) = floor((5n+7)/12) - floor((5n+2)/12). - David W. Wilson, Mar 02 2003
G.f.: -x*(1+x^3+x^5+x^8+x^10) / ( (x-1) *(1+x+x^2) *(1+x) *(1-x+x^2) *(1+x^2) *(x^4-x^2+1) ). - R. J. Mathar, Dec 16 2016
a(n) = 0 if n is in A060107, a(n) = 1 if n is in A060106. - Jianing Song, Oct 14 2019

A319451 Numbers that are congruent to {0, 3, 6} mod 12; a(n) = 3floor(4n/3).

Original entry on oeis.org

0, 3, 6, 12, 15, 18, 24, 27, 30, 36, 39, 42, 48, 51, 54, 60, 63, 66, 72, 75, 78, 84, 87, 90, 96, 99, 102, 108, 111, 114, 120, 123, 126, 132, 135, 138, 144, 147, 150, 156, 159, 162, 168, 171, 174, 180, 183, 186, 192, 195, 198, 204, 207, 210, 216, 219, 222, 228

Offset: 0

Jianing Song, Sep 19 2018

Key-numbers of the pitches of a diminished chord on a standard chromatic keyboard, with root = 0.

Jianing Song, Table of n, a(n) for n = 0..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,1,-1).

Cf. A004773, A047464.
A guide for some sequences related to modes and chords:
Modes:
Lydian mode (F): A083089
Ionian mode (C): A083026
Mixolydian mode (G): A083120
Dorian mode (D): A083033
Aeolian mode (A): A060107 (raised seventh: A083028)
Phrygian mode (E): A083034
Locrian mode (B): A082977
Third chords:
Major chord (F,C,G): A083030
Minor chord (D,A,E): A083031
Diminished chord (B): this sequence
Seventh chords:
Major seventh chord (F,C): A319280
Dominant seventh chord (G): A083032
Minor seventh chord (D,A,E): A319279
Half-diminished seventh chord (B): A319452

Filtered([0..230],n->n mod 12 = 0 or n mod 12 = 3 or n mod 12 = 6); # Muniru A Asiru, Oct 24 2018

[n : n in [0..150] | n mod 12 in [0, 3, 6]]

seq(3*floor(4*n/3),n=0..60); # Muniru A Asiru, Oct 24 2018

Select[Range[0, 200], MemberQ[{0, 3, 6}, Mod[#, 12]]&]
LinearRecurrence[{1, 0, 1, -1}, {0, 3, 6, 12}, 100]
Table[4n-1+Sin[Pi/3(2n+1)]/Sin[Pi/3],{n,0,99}] (* Federico Provvedi, Oct 23 2018 *)

PARI
```
a(n)=3*(4*n\3)
```

for n in range(0,60): print(3*int(4*n/3), end=", ") # Stefano Spezia, Dec 07 2018

a(n) = a(n-3) + 12 for n > 2.
a(n) = a(n-1) + a(n-3) - a(n-4) for n > 3.
G.f.: 3*(1 + x + 2*x^2)/((1 - x)*(1 - x^3)).
a(n) = 3*A004773(n) = 3*(floor(n/3) + n).
a(n) = 4*n - 1 + sin((Pi/3)*(2*n + 1))/sin(Pi/3). - Federico Provvedi, Oct 23 2018
E.g.f.: (3*cos(sqrt(3)*x/2) + sqrt(3)*sin(sqrt(3)*x/2))/(3*exp(x/2)) - exp(x)*(1 - 4*x). - Franck Maminirina Ramaharo, Nov 27 2018
Sum_{n>=2} (-1)^n/a(n) = (sqrt(2)-1)*Pi/24 + (2-sqrt(2))*log(2)/24 + sqrt(2)*log(2+sqrt(2))/12. - Amiram Eldar, Dec 30 2021

A060106 Numbers that are congruent to {1, 4, 6, 9, 11} mod 12. The ebony keys on a piano, starting with A0 = the 0th key.

Original entry on oeis.org

1, 4, 6, 9, 11, 13, 16, 18, 21, 23, 25, 28, 30, 33, 35, 37, 40, 42, 45, 47, 49, 52, 54, 57, 59, 61, 64, 66, 69, 71, 73, 76, 78, 81, 83, 85, 88, 90, 93, 95, 97, 100, 102, 105, 107, 109, 112, 114, 117, 119, 121, 124, 126, 129, 131, 133, 136, 138, 141, 143, 145, 148

Offset: 1

Henry Bottomley, Feb 27 2001

A piano sequence since if a(n) < 88 then A059620(a(n)) = 1.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,1,-1).

Cf. A059620, A081032. Complement of A060107.

Vec(x*(1 + 3*x + 2*x^2 + 3*x^3 + 2*x^4 + x^5) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)) + O(x^60)) \\ Colin Barker, Oct 14 2019

a(n) = a(n-5) + 12.
a(n) = A081032(n) - 1 for 1 <= n <= 36. - Jianing Song, Oct 14 2019
From Colin Barker, Oct 14 2019: (Start)
G.f.: x*(1 + 3*x + 2*x^2 + 3*x^3 + 2*x^4 + x^5) / ((1 - x)^2*(1 + x + x^2 + x^3 + x^4)).
a(n) = a(n-1) + a(n-5) - a(n-6) for n > 6.
(End)

A319279 Numbers that are congruent to {0, 3, 7, 10} mod 12.

Original entry on oeis.org

0, 3, 7, 10, 12, 15, 19, 22, 24, 27, 31, 34, 36, 39, 43, 46, 48, 51, 55, 58, 60, 63, 67, 70, 72, 75, 79, 82, 84, 87, 91, 94, 96, 99, 103, 106, 108, 111, 115, 118, 120, 123, 127, 130, 132, 135, 139, 142, 144, 147, 151, 154, 156, 159, 163, 166, 168, 171, 175, 178

Offset: 1

Jianing Song, Sep 16 2018

Key-numbers of the pitches of a minor seventh chord on a standard chromatic keyboard, with root = 0.

Apart from the offset the same as A013574. - R. J. Mathar, Sep 27 2018

Jianing Song, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-1).

A guide for some sequences related to modes and chords:
Modes:
Lydian mode (F): A083089
Ionian mode (C): A083026
Mixolydian mode (G): A083120
Dorian mode (D): A083033
Aeolian mode (A): A060107 (raised seventh: A083028)
Phrygian mode (E): A083034
Locrian mode (B): A082977
Third chords:
Major chord (F,C,G): A083030
Minor chord (D,A,E): A083031
Diminished chord (B): A319451
Seventh chords:
Major seventh chord (F,C): A319280
Dominant seventh chord (G): A083032
Minor seventh chord (D,A,E): this sequence
Half-diminished seventh chord (B): A319452

[n : n in [0..150] | n mod 12 in [0, 3, 7, 10]]

Select[Range[0, 200], MemberQ[{0, 3, 7, 10}, Mod[#, 12]]&]
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 3, 7, 10, 12}, 100]

x='x+O('x^99); concat(0, Vec(x^2*(3+x+2*x^2)/((x^2+1)*(x-1)^2)))

a(n) = a(n-4) + 12 for n > 4.
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
G.f.: x^2*(3 + x + 2*x^2)/((x^2 + 1)*(x - 1)^2).
a(n) = (6*n - 5 + sqrt(2)*cos(Pi*n/2 + Pi/4))/2.
E.g.f.: ((6x - 5)*e^x + sqrt(2)*cos(x + Pi/4) + 4)/2.

A081031 Positions of white keys on piano keyboard, starting with A0 = the 1st key.

Original entry on oeis.org

1, 3, 4, 6, 8, 9, 11, 13, 15, 16, 18, 20, 21, 23, 25, 27, 28, 30, 32, 33, 35, 37, 39, 40, 42, 44, 45, 47, 49, 51, 52, 54, 56, 57, 59, 61, 63, 64, 66, 68, 69, 71, 73, 75, 76, 78, 80, 81, 83, 85, 87, 88

Offset: 1

David W. Wilson, Mar 02 2003

			First, 3rd, 4th, 6th, etc. keys of piano keyboard are white.

Robbert Fokkink, The Pell Tower and Ostronometry, arXiv:2309.01644 [math.CO], 2023.
Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,1,-1).

Cf. A059620, A081032, A060107.

Join[{1,3},Flatten[Table[12n+{4,6,8,9,11,13,15},{n,0,6}]],{88}] (* Harvey P. Dale, Mar 15 2013 *)
LinearRecurrence[{1,0,0,0,0,0,1,-1},{1,3,4,6,8,9,11,13},52] (* Harvey P. Dale, May 14 2023 *)

a(n) = floor((12n-3)/7).
From Chai Wah Wu, Sep 11 2018: (Start)
a(n) = a(n-1) + a(n-7) - a(n-8) for n > 8.
G.f. for a keyboard with infinite number of keys: x*(x^7 + 2*x^6 + x^5 + 2*x^4 + 2*x^3 + x^2 + 2*x + 1)/(x^8 - x^7 - x + 1). (End)
a(n) = A060107(n) + 1 for 1 <= n <= 36. - Jianing Song, Oct 14 2019

A319280 Numbers that are congruent to {0, 4, 7, 11} mod 12.

Original entry on oeis.org

0, 4, 7, 11, 12, 16, 19, 23, 24, 28, 31, 35, 36, 40, 43, 47, 48, 52, 55, 59, 60, 64, 67, 71, 72, 76, 79, 83, 84, 88, 91, 95, 96, 100, 103, 107, 108, 112, 115, 119, 120, 124, 127, 131, 132, 136, 139, 143, 144, 148, 151, 155, 156, 160, 163, 167, 168, 172, 175, 179

Offset: 1

Jianing Song, Sep 16 2018

Key-numbers of the pitches of a major seventh chord on a standard chromatic keyboard, with root = 0.

Jianing Song, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

A guide for some sequences related to modes and chords:
Modes:
Lydian mode (F): A083089
Ionian mode (C): A083026
Mixolydian mode (G): A083120
Dorian mode (D): A083033
Aeolian mode (A): A060107 (raised seventh: A083028)
Phrygian mode (E): A083034
Locrian mode (B): A082977
Third chords:
Major chord (F,C,G): A083030
Minor chord (D,A,E): A083031
Diminished chord (B): A319451
Seventh chords:
Major seventh chord (F,C): this sequence
Dominant seventh chord (G): A083032
Minor seventh chord (D,A,E): A319279
Half-diminished seventh chord (B): A319452

[n : n in [0..150] | n mod 12 in [0, 4, 7, 11]];

Select[Range[0, 200], MemberQ[{0, 4, 7, 11}, Mod[#, 12]]&]
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 4, 7, 11, 12}, 100]

my(x='x+O('x^99)); concat(0, Vec(x^2*(4+3*x+4*x^2+x^3)/((1+x)*(1+x^2)*(1-x)^2)))

a(n) = a(n-4) + 12 for n > 4.
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
G.f.: x^2*(4 + 3*x + 4*x^2 + x^3)/((1 + x)*(1 + x^2)*(1 - x)^2).
a(n) = (6*n - 4 + (-1)^n + sqrt(2)*cos(Pi*n/2 + Pi/4))/2.
E.g.f.: ((6*x - 3)*cosh(x) + (6*x - 5)*sinh(x) + sqrt(2)*cos(x + Pi/4) + 2)/2.
Sum_{n>=2} (-1)^n/a(n) = log(3)/8 + log(2+sqrt(3))/(2*sqrt(3)) - 5*sqrt(3)*Pi/72. - Amiram Eldar, Dec 30 2021

A319452 Numbers that are congruent to {0, 3, 6, 10} mod 12.

Original entry on oeis.org

0, 3, 6, 10, 12, 15, 18, 22, 24, 27, 30, 34, 36, 39, 42, 46, 48, 51, 54, 58, 60, 63, 66, 70, 72, 75, 78, 82, 84, 87, 90, 94, 96, 99, 102, 106, 108, 111, 114, 118, 120, 123, 126, 130, 132, 135, 138, 142, 144, 147, 150, 154, 156, 159, 162, 166, 168, 171, 174, 178

Offset: 1

Jianing Song, Sep 19 2018

Key-numbers of the pitches of a half-diminished chord on a standard chromatic keyboard, with root = 0.

Jianing Song, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (1,0,0,1,-1).

A guide for some sequences related to modes and chords:
Modes:
Lydian mode (F): A083089
Ionian mode (C): A083026
Mixolydian mode (G): A083120
Dorian mode (D): A083033
Aeolian mode (A): A060107 (raised seventh: A083028)
Phrygian mode (E): A083034
Locrian mode (B): A082977
Third chords:
Major chord (F,C,G): A083030
Minor chord (D,A,E): A083031
Diminished chord (B): A319451
Seventh chords:
Major seventh chord (F,C): A319280
Dominant seventh chord (G): A083032
Minor seventh chord (D,A,E): A319279
Half-diminished seventh chord (B): this sequence

[n : n in [0..150] | n mod 12 in [0, 3, 6, 10]]

Select[Range[0, 200], MemberQ[{0, 3, 6, 10}, Mod[#, 12]]&]
LinearRecurrence[{1, 0, 0, 1, -1}, {0, 3, 6, 10, 12}, 100]

my(x='x+O('x^99)); concat(0, Vec(x^2*(3+3*x+4*x^2+2*x^3)/((1+x)*(1+x^2)*(1-x)^2)))

a(n) = a(n-4) + 12 for n > 4.
a(n) = a(n-1) + a(n-4) - a(n-5) for n > 5.
G.f.: x^2*(3 + 3*x + 4*x^2 + 2*x^3)/((1 + x)*(1 + x^2)*(1 - x)^2).
a(n) = (12*n - 11 + (-1)^n + 2*cos(Pi*n/2))/4.
E.g.f.: ((6*x - 5)*cosh(x) + (6*x - 6)*sinh(x) + cos(x) + 4)/2.
Sum_{n>=2} (-1)^n/a(n) = log(12)/8 - (sqrt(3)-1)*Pi/24. - Amiram Eldar, Dec 30 2021

A356464 Number of black keys in each group of black keys on a standard 88-key piano (left to right).

Original entry on oeis.org

1, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3

Offset: 1

Peter Woodward, Aug 08 2022

On a standard piano keyboard, the black keys appear in groups of two and three, with each group separated from adjacent groups by the presence of two white keys that have no black key between them.
The black keys in a group of two are C#/Db and D#/Eb; the black keys in a group of three are F#/Gb, G#/Ab, and A#/Bb.
The A#/Bb key near the left end of the keyboard is a special case; it is the only black key in its group because the white A key to its left is the leftmost key on the keyboard.

			From _Jon E. Schoenfield_, Aug 12 2022: (Start)
In the diagram below, five octaves (i.e., sets of 12 consecutive keys) have been omitted (as represented by the ellipses):
.
    n |  1       2         3       ...     14        15
  ----+---------------------------------------------------------
  a(n)|  1       2         3       ...      2         3
    ______________________________ ... _________________________
      | |/| | |/||/| | |/||/||/| |     | |/||/| | |/||/||/| |  |
      | |/| | |/||/| | |/||/||/| |     | |/||/| | |/||/||/| |  |
      | |/| | |/||/| | |/||/||/| |     | |/||/| | |/||/||/| |  |
      | |_| | |_||_| | |_||_||_| |     | |_||_| | |_||_||_| |  |
      |  |  |  |  |  |  |  |  |  |     |  |  |  |  |  |  |  |  |
      |  |  |  |  |  |  |  |  |  |     |  |  |  |  |  |  |  |  |
      |__|__|__|__|__|__|__|__|__|     |__|__|__|__|__|__|__|__|
       A  B  C  D  E  F  G  A  B   ...  C  D  E  F  G  A  B  C
(End)

Cf. A059620, A060107, A060106, A081031, A081032.

Cf. A329207.

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A083120 Numbers that are congruent to {0, 2, 4, 5, 7, 9, 10} mod 12.

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A059620 Colors of the 88 keys of the standard piano: white keys = 0, black keys = 1, start with A0 = the 0th key.

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A319451 Numbers that are congruent to {0, 3, 6} mod 12; a(n) = 3*floor(4*n/3).

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A060106 Numbers that are congruent to {1, 4, 6, 9, 11} mod 12. The ebony keys on a piano, starting with A0 = the 0th key.

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A319279 Numbers that are congruent to {0, 3, 7, 10} mod 12.

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A081031 Positions of white keys on piano keyboard, starting with A0 = the 1st key.

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A319280 Numbers that are congruent to {0, 4, 7, 11} mod 12.

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A319451 Numbers that are congruent to {0, 3, 6} mod 12; a(n) = 3floor(4n/3).